have been from the infancy of culture, may be well shown by the permanence of the practice of holding at intervals such special ceremonies to expel them. In Siam the people first hunt the demons out of the houses, and then drive them with cannon-shots through the streets till they get them outside the walls into the forest. In Old Calabar they put puppets along the streets leading to the sea, to entice the demons into, and then at dead of night a sudden rush is made by the negroes with whips and torches to drive the spirits down into the sea. Not only do other barbaric regions, such as the South Sea Islands and Peru, furnish similar examples of the expulsion of demons, but it may still be seen among European peasantry. In Sweden, Easter-tide is the season for a general purging of the land from the evil spirits and trolls of the old heathendom; and in many parts of Germany unseen witches are to this day driven out on Walpurgisnight with crack of whip and blast of horn. (See a collection of cases in Bastian and Hartmann, Zeitschrift für Ethnologie, 1869, p. 189; also Hylten-Cavallius, Wärend och Wirdarne, part i. p. 178). In these cases it is generally unfavourable influences which are considered as due to the demons. But favourable events are even by savages often recognized as due to the intervention of some kindly spirit, and especially to a guardian or patron demon, whose help accounts for what among ourselves is often not much more rationally considered to be "luck." It is often a recognized ancestral soul which from natural affection undertakes this duty, as when a Tasmanian has been known to account for escape from danger by the idea that his father's soul was still watching over him. But it need not be so; and among the American Indians or West Africans, where each man lives in constant imaginary intercourse with his patron-spirit, talking with it, making it offerings, and trusting to its guidance in difficulty and protection from danger, this spirit may be revealed in a dream or vision, and is often connected with some object known as a "medicine" or "fetish," but is seldom identified with any particular ghost. In Greek literature this idea is best exemplified by the lines of Menander on the good demon whom every man has from birth as his guide through the mysteries of life (ap. Clem. Alex., Stromat. v.); the most popularly known example is the so-called "demon of Socrates, but he himself did not give such personal definiteness to the divine or dæmonic influence (Sauóviov) which warned him by what he described as a voice or sign (see Zeller, Socrates, ch. 4). The primitive idea of the patron spirit is carried on in the Roman genius, whose name (even without the addition of "natalis ") indicates that it is born with the person whom it accompanies through life. Its place very closely corresponds to that occupied in modern folklore by the guardian angel. There are districts in France where a peasant meeting another, salutes not only the man, but his "companion," the guardian angel who is supposed to be invisibly at his side.

Among attendant and patron demons, as recognized in the general belief of mankind, a specially important class is formed by the familiar spirits who accompany sorcerers, giving them mysterious knowledge, uttering oracular responses through their voices, enabling them to perform wonderful feats, bringing them treasure or injuring their nemies, and doing other spiritual services for them. From the descriptions of sorcerers among the lower nations, it is at once evident that their supposed intercourse with demons is closely connected with the symptoms of diseasepossession. Thus among the Zulus, "the disease which precedes the power to divine" is distinctly hysterical, the patient's morbid sensitiveness and intensely vivid imagination of sights and voices fitting well with his persuasion that he is under the control of some ancestral ghost. So

well is this connection recognized among races like the Patagonians and rude tribes of Siberia, that children with an hereditary tendency to epilepsy are brought up to the profession of magicians. Where the sorcerer has not naturally such symptoms of possession by a controlling demon, he is apt to bring them on by violent dancing and beating drums, or by drugs, or to simulate them by mere kuavery; which latter is really the most convincing proof that the original notion of the demon of the magician did not arise from imposture, but from actual belief that the morbid excitement, hallucination, and raving consequent on mental disease were caused by spirits other than tho man's own soul, in possession of his body. The primitive and savage theory of inspiration by another spirit getting inside the body is most materialistic, and cheating sorcerers accordingly use ventriloquism of the original kind, which (as its name implies) is supposed to be caused by the voice of a demon inside the body of the speaker, who really himself talks in a feigned human voice, or in squeaking or whistling tones thought suitable to the thin-bodied spirit-visitor. The familiar spirit may be a human ghost or some other demon, and may either be supposed to enter the man's body or only to come into his presence, which is somewhat the same difference as whether in disease the demon possesses" or "obsesses a patient, i.e., controls him from inside or outside. Thus the Greenland angekok, or sorcerer, is described as following his profession by the aid of a torngak, or familiar spirit (who may be an ancestral ghost), whom he summous by drumming, and with whom he is heard by the bystanders to carry on a conversation within the hut, obtaining information which enables him to advise as to the treatment of the sick, the prospect of good or bad weather, and the other topics of the business of a soothsayer. Passing over the intermediate space which divides the condition of savages from that of medieval or modern Europeans, we shall find, so far as the doctrine of familiar demons has survived, that it has changed but little in principle. In the witch trials a favourite accusation was that of having a familiar demon. Sir Walter Scott's Demonology and Witchcraft contains among others the case of Bessie Dunlop, whose familiar was the ghost of one Thome Reid, killed at the battle of Pinkie (1547), who enabled her to give answers to such as consulted her about the ailments of human beings or cattle, or the recovery of things lost or stolen. This miserable woman, chiefly on her own confession, was as usual "convict and burnt." Here the imagined demon was a human soul; but other spirits thus attended sorcerers and diviners, such as the spirit called Hudhart, who enabled a certain Highland woman to prophesy as to the conspiracy to murder James I. of Scotland. Dissertations on the art

of raising demons for the sorcerers' service, and even the actual charms and ceremonies to be used, form a large part of the precepts of magical books. (See Ennemoser, History of Magic; Horst, Zauberbibliothek, and other works already cited.) Among the latest English books treating seriously of this "black art" is Sibly's Illustration of the Occult Sciences, of which a 10th edition, in 4to, bears date London, 1807. The statute of James L. of England enacts that all persons invoking any evil spirit, or consulting, covenanting with, entertaining, employing, feeding, or rewarding any evil spirit, should be guilty of felony, and suffer death. This was not repealed till the reign of George II. Educated public opinion has now risen above this level; but popular credulity is still to be worked upon by much the same means as those employed by savage sorcerers professing intercourse with familiar spirits. At "spiritualistic séances" the convulsive and hysterical symptoms (pretended or real) of the "medium" under the "control" of his "guiding spirit are much the same as those which

teen years had learned Latin, Greek, and some Hebrew, in addition to acquiring much general knowledge. At the age of sixteen years and a half he entered Trinity College, Cambridge, and studied mathematics. partly under the tuition of Airy, subsequently the astronomer royal. In 1825 he gained a Trinity scholarship. De Morgan's attention was by no means confined to mathematics, and his love of wide reading somewhat interfered with his success in the mathematical tripos, in which he took the fourth place in 1827, before he had completed his twenty-first year. He was prevented from taking his M. A. degree, or from obtaining a fellowship, to which he would doubtless have been elected, by his conscientious objection to signing the theological tests then required from masters of arts and fellows at Cambridge. A strong repugnance to any sectarian restraints upon the freedom of opinion was one of De Morgan's most marked characteristics throughout life. A career in his own university being closed against him, he entered Lincoln's Inn; but had hardly done so when the establishment, in 1828, of the university of London, in Gower Street, afterwards known as University College, gave him an opportunity of continuing his mathematical pursuits. At the early age of twenty-two years he gave his first lecture as professor of mathematics in a college which he served with the utmost zeal and success for a third of a century. His connection with the college, indeed, was interrupted in 1831, when a disagreement with the governing body caused De Morgan and some other professors to resign their chairs simultaneously. When, in 1836, his successor Mr White was accidentally drowned, De Morgan was requested to resume the rrofessorship. It may be added that his choice of a literary and scientific career was made against the advice of his relatives and friends, who, on his entering Lincoln's Inn, confidently anticipated for him a distinguished and lucrative career at

the bar.

In 1837 De Morgan married Sophia Elizabeth, daughter of William Frend, a Unitarian in faith, a mathematician and actuary in occupation, a notice of whose life, written by his son-in-law, will be found in the Monthly Notices of the Royal Astronomical Society (vol. v). Henceforward De Morgan's life is scarcely more than a record of his constant labours, and his innumerable publications. As in the case of many scholars, the even tenor of his life was unbroken by remarkable incidents. Surrounded by a growing family, ultimately seven in number, he sought happiness in his home, in his library, and in the energetic and vigorcus discharge of his college duties. He seldom travelled or enjoyed relaxation, and could with difficulty be induced to remain many days from home.

As a teacher of mathematics De Morgan was unrivalled. He gave instruction in the form of continuous lectures delivered extempore from brief notes. The most prolonged mathematical reasoning, and the most intricate formulae, were given with almost infallible accuracy from the resources of his extraordinary memory. De Morgan's writings, however excellent, give little idea of the perspicuity and elegance of his viva voce expositions, which never failed to fix the attention of all who were worthy of hearing him. Many of his pupils have distinguished themselves, and, through Mr Todhunter and Mr Routh, he has had an important influence on the modern Cambridge school. In addition to occasional extra courses, it was his habit to give two lectures on each of the six week days throughout the working session of thirty weeks or more. Each lecture was exactly one hour and a quarter in length, and at the close amber of questions and problems were always given, to which the pupils returned written answers. These were all corrected by the professor's own hand, and personal explanations given before or after the lecture.

Although the best hours of the day were thus given to arduous college work, his public labours in other directions were extensive. For thirty years he took an active part in the business of the Royal Astronomical Society, editing its publications, supplying obituary notices of members, and for 18 years acting as one of the honorary secretaries. His work for this society alone, it is said, would have been occupation enough for an ordinary man. He was also frequently employed as consulting actuary, a business in which his mathematical powers, combined with sound judgment and business-like habits, fitted him to take the highest place.

De Morgan's mathematical writings contributed powerfully towards the progress of the science. His memoirs on the "Foundation of Algebra," in the 7th and 8th volumes of the Cambridge Philosophical Transactions, contain some of the most important contributions which have been made to the philosophy of mathematical method; and Sir W. Rowan Hamilton, in the preface to his Lectures on Quaternions, refers more than once to those papers as having led and encouraged him in the working out of the new system of quaternions. The work on Trigonometry and Double Algebra, published by De Morgan in 1849, contains in the latter part a most luminous and philosophical view of existing and possible systems of symbolic calculus. But De Morgan's influence on mathematical science in England can only be estimated by a review of his long series of publications, which cominence, in 1828, with a translation of part of Bourdon's Elements of Algebra, prepared for his students. In 1830 appeared the first edition of his well-known Elements of Arithmetic, which has been widely used in schools, and has done much to raise the character of elementary training. It is distinguished by a simple yet thoroughly philosophical treatment of the ideas of number and magnitude, as well as by the introduction of new abbreviated processes of computation, to which De Morgan always attributed much practical importance. Second and third editions were called for in 1832 and 1835, and more than 20,000 copies have been sold; the book is still in use, a sixth edition having been issued in 1876.

De Morgan's other principal mathematical works were The Elements of Algebra, 1835, a valuable but somewhat dry elementary treatise; the Essay on Probabilities, 1838, forming the 107th volume of Lardner's Cyclopædia, still much used, being probably the best simple introduction to the theory in the English language; and The Elements of Trigonometry and Trigonometrical Analysis, preliminary to the Differential Calculus, 1837.

Several of his mathematical works were published by the Society for the Diffusion of Useful Knowledge, of which De Morgan was at one time an active member. Among these may be mentioned the great Treatise on the Differential and Integral Calculus, 1842, which still remains the most extensive and complete English treatise on the subject; the Elementary Illustrations of the Differential and Integral Calculus, first published in 1832, but often bound up with the larger treatise; the valuable essay, On the Study and Difficulties of Mathematics, 1831; and a brief treatise on Spherical Trigonometry, 1834. By some accident the work on probability in the same series, written by Lubbock and DrinkwaterBethune was attributed to De Morgan, an error which seriously annoyed his nice sense of bibliographical accuracy. For fifteen years he did all in his power to correct the mistake, and finally wrote to the Times to disclaim the authorship. (See Monthly Notices of the Royal Astronomical Society, vol. xxvi. p. 118.)

Two of his most elaborate treatises are to be found in the Encyclopædia Metropolitana, namely the articles on the

VII.

trations, and abundant proofs of De Morgan's varied learning. Unfortunately these memoirs are accessible to few readers, otherwise they would form invaluable reading for the logical student. In 1860 De Morgan eudeavoured to render their contents better known by publishing a Syllabus of a Proposed System of Logic, from which may be obtained a good idea of his symbolic system, but the more readable and interesting discussions contained in the memoirs are of necessity omitted. The article "Logic" in the English Cyclopædia (1860) completes the list of his logical publications.

In spite of the excellence and extent of his mathematical writings, it is probably as a logical reformer that De Morgan will be best known to future times. In this respect he stands alongside of his great contemporaries Hamilton and Boole, as one of several independent discover ers of the all-important principle of the quantification of the predicate. Unlike most mathematicians, De Morgan always laid much stress upon the importance of logical training. In his admirable papers upon the modes of teaching arithmetic and geometry, originally published in the Quarterly Journal of Education (reprinted in The Schoolmaster, vol ii.), he remonstrated against the neglect of logical doctrine. In 1839 he produced a small work called First Notions of Logic, giving what he had found by experience to be much wanted by students commencing with Euclid.

In October 1846 he completed the first of his original investigations, in the form of a paper printed in the Transactions of the Cambridge Philosophical Society (vol. viii. No. 29). In this paper the principle of the quantified predicate was referred to, and there immediately ensued a memorable controversy with Sir W. Hamilton regarding the independence of De Morgan's discovery, some communications having passed between them in the autumn of 1846. The details of this dispute will be found by those interested in the original pamphlets, in the Athenæum newspaper, or in the appendix to De Morgan's Formal Logic. Suffice it to say that the independence of De Morgan's discovery was subsequently recognized by Hamilton, and that those acquainted with De Morgan's character could never suppose that it was otherwise. Moreover, the eight forms of proposition adopted by De Morgan as the basis of his system partially differ from those which Hamilton derived from the quantified predicate. The general character of De Morgan's development of logical forms was wholly peculiar and original on his part.

Not a year passed before De Morgan, late in 1847, published his principal logical treatise, called Formal Logic, or the Calculus of Inference, Necessary and Probable. This contains a reprint of the First Notions, an elaborate development of his doctrine of the syllogism, and of the numerically definite syllogism, together with chapters of great interest on probability, induction, old logical terms, and fallacies. The severity of the treatise is relieved by characteristic touches of humour, and by quaint anecdotes and allusions furnished from his wide reading and perfect

memory.

There followed at intervals, in the years 1850, 1858, 1860, and 1863, a series of four elaborate memoirs on the "Syllogism," printed in volumes ix. and x. of the Cambridge Philosophical Transactions. These papers taken together constitute a great treatise on logic, in which he substituted improved systems of notation, and developed a new logic of relations, and a new onymatic system of logical expression. Apart, however, from their principal purpose,

Throughout his logical writings De Morgan was led by the idea that the followers of the two great branches of exact science, logic and mathematics, had made blunders,— the logicians in neglecting mathematics, and the mathematicians in neglecting logic. He endeavoured to reconcile them, and in the attempt showed how many errors an acute mathematician could detect in logical writings; and how large a field there was for discovery. But it may be doubted whether De Morgan's own system, "horrent with mysterious spiculæ," as Hamilton aptly described it, is fitted to exhibit the real analogy between quantitative and qualitative reasoning, which is rather to be sought in the logical works of Boole. (See BOOLE, vol. iv. p. 47.)

Perhaps the largest part, in volume, of De Morgan's writings remains still to be briefly mentioned; it consists of detached articles contributed to various periodical or composite works. During the the Penny Cyclopædia, writing chiefly on mathematics, astronomy, years 1833-43, he contributed very largely to the first edition of physics, and biography. His articles of various length cannot be less in number than 850, as may be ascertained from a signed copy in the British Museum, and they have been estimated to constitute a sixth part of the whole Cyclopædia, of which they formed perhaps the most valuable portion. He also wrote biographies of Newton and Halley for Knight's British Worthics, various notices of scientific men for the Gallery of Portraits, and for the uncompleted Biographical Dictionary of the Useful Knowledge Society, and at Biography. least seven articles in Smith's Dictionary of Greek and Roman

Some of De Morgan's most interesting and useful minor writings are to be found in the Companions to the British Almanack, to which he contributed without fail one article each year from 1831 up to variety of topics relating to astronomy, chronology, decimal coinage, 1857 inclusive. In these carefully written papers he treats a great life-assurance, bibliography, and the history of science. Most of them are as valuable now as when written.

Among De Morgan's miscellaneous writings may be mentioned including a description of the maps of the stars, published by the his Explanation of the Gnomonic Projection of the Sphere, 1836, Useful Knowledge Society; his Treatise on the Globes, Celestial and Terrestrial, 1845; and his remarkable Book of Almanacks, (second edition 1871), which contains a series of 35 almanacks, so arranged with indices of reference, that the almanack for any year, whether in old style or new, from any epoch, ancient or modern, up to 2000 A.D., may be found without difficulty, means being added for verifying the almanack and also for discovering the days of new and full moon from 2000 B.c. up to 2000 A.D. D Morgan expressly draws attention to the fact that the plan of this book was that of Francœur and Ferguson, but the plan was developed by one who was an unrivalled master of all the intricacies of chronology. The two best tables of logarithms, the small fivefigure tables of the Useful Knowledge Society (1839 and 1857), and De Morgan's superintendence. Several works edited by him will Shroen's Seven Figure-Table (5th ed. 1865), were printed under be found mentioned in the British Museum Catalogue. His numerous anonymous contributions through a long series of years to the Athenæum, and to Notes and Queries, and his occasional must be passed over with this bare mention. articles in the North British Review, Macmillan's Magazine, &c.,

Considerable labour was spent by De Morgan upon the subject of decimal money. He was a great advocate of the pound and mil scheme. His evidence on this subject was sought by the Royal Commission, and, besides constantly supporting the Decimal Association in periodical publications, he published several separate pamphlets on the subject.

One marked character of De Morgan was his intense and yet reasonable love of books. He was a true bibliophil, and loved to surround himself, as far as his means allowed, with curious and rare books. He revelled in all the mysteries of watermarks, title pages, colophons, catch-words, and the like; yet he treated biblio

graphy as an important science. As he himself wrote, "the most worthless book of a bygone day is a record worthy of preservation; like a telescopic star, its obscurity may render it unavailable for most-purposes; but it serves, in hands which know how to use it, to determine the places of more important bodies." His evidence before the Royal Commission on the British Museum in 1850, (Questions 5704*-5815,* 6481-6513, and 8966-8967), should be studied by all who would comprehend the principles of bibliography or the art of constructing a catalogue, his views on the latter subject corresponding with those carried out by Panizzi in the British Museum Catalogue. A sample of De Morgan's bibliographical learning is to be found in his account of Arithmetical Books, from the Invention of Printing (1847), and finally in his Budget of Paradoxes. This latter work consists of articles most of which were originally published in the Athenæum, describing the various attempts which have been made to invent a perpetual motion, to square the circle, or to trisect the angle; but De Morgan took the opportunity to include many curious bits gathered from his extensive reading, so that the Budget as reprinted by his widow (1872), with much additional matter prepared by himself, forms a remarkable collection of scientific ana. De Morgan's correspondence with contemporary scientific men was very extensive and full of interest. It remains unpublished, as does also a large mass of mathematical tracts which he prepared for the use of his students, treating all parts of mathematical science, and embodying some of the matter of his lectures. De Morgan's library was purchased by Lord Overstone, and presented to the university of London.

From the above enumeration it will be apparent that the extent of De Morgan's literary and scientific labours was altogether extraordinary; nor was quality sacrificed to quantity. On the contrary every publication was finished with extreme care and accuracy, and no writer can be more safely trusted in every thing which he wrote. It is possible that his continual efforts to attain completeness and absolute correctness injured his literary style, which is wanting in grace; but the estimation in which his books are held is shown by the fact that they are steadily rising in market price. Apart from his conspicuous position as a logical and mathematical discoverer, we may conclude that hardly any man of science in recent times has had a more extensive, though it may often be an unfelt influence, upon the progress of exact and sound knowledge.1

De Morgan has left no published indications of his opinions on religious questions, in regard to which he was extremely reticent. He seldom or never entered a place of worship, and declared that he could not listen to a sermon, a circumstance perhaps due to the extremely strict religious discipline under which he was brought up. Nevertheless there is reason to believe that he was of a deeply religious Cisposition. Like Faraday and Newton he entertained a confident belief in Providence, founded not on any tenuous method of inference, but on personal feeling. His hope of a future life also was vivid to the last.

In the year 1866 a life as yet comparatively free from trouble became clouded by the circumstances which led him to abandon the institution so long the scene of his labours. The refusal of the council to accept the recommendation of the senate, that they should appoint an eminent Unitarian minister to the professorship of logic and mental philosophy, revived all De Morgan's sensitiveness on the subject of sectarian freedom; and, though his feelings were doubtless excessive, there is no doubt that gloom was thrown over his life, intensified in 1867 by the loss of his son George Campbell De Morgan, a young man of the highest scientific promise, whose name, as De Morgan expressly wished, will long be connected with the London Mathematical Society, of which he was one of the founders. From this time De Morgan rapidly fell into ill-health, previously almost

In a notice of De Morgan's character it is impossible to omit a reference to his witty sayings, some specimens of which are preserved in Dr Sadler's most interesting Diary of Henry Crabb Robinson (1869), which also contains a humorous account of H. C. R. by De Morgan. It may be added that De Morgan was a great reader and admirer of Dickens; he was also fond of music, and a fair performer on the fate

unknown to him, dying on the 18th March 1871. An interesting and truthful sketch of his life will be found in the Monthly Notices of the Royal Astronomical Society, for the 9th February 1872, vol. xxii. p. 112, written by Mr Ranyard, who says, "He was the kindliest, as well as the most learned of men-benignant to every one who approached him, never forgetting the claims which weakness has on strength." (W. S. J.)

DEMOSTHENES was born in 384 B.C. His father, who bore the same name, was an Athenian citizen belonging to the deme of Pæania. His mother, Cleobule, was the daughter of Gylon, a citizen who had been active in procuring the protection of the kings of Bosporus for the Athenian colony of Nymphæon in the Crimea, and whose wife was a native of that region. On these grounds the adversaries of Demosthenes, in after-days, used absurdly to taunt him with a traitorous or barbarian ancestry. The boy had a bitter foretaste of life. He was seven years old when his father died, leaving property (in a manufactory of swords, and another of upholstery) worth about £3500, which, invested as it seems to have been (20 per cent. was not thought exorbitant), would have yielded rather more than £600 a year. £300 a year was a very comfortable income at Athens, and it was possible to live decently on a tenth of it. Nicias, a very rich man, had property equivalent, probably, to not more than £4000 a year. Demosthenes was born, then, to a handsome, though not a great fortune. But his guardians—two nephews of his father, Aphobus and Demophon, and one Therippidesabused their trust, and handed over to Demosthenes, when he came of age, rather less than one-seventh of his patrimony, perhaps between £50 and £60 a year. Demosthenes, after studying with Isæus-then the great master of forensic eloquence and of Attic law, especially in will cases 2-brought an action against Aphobus, and gained a verdict for about £2400. But it does not appear that he got the money; and, after some more fruitless proceedings against Onetor, the brother-in-law of Aphobus, the matter was dropped,-not, however, before his relatives had managed to throw a public burden (the equipment of a ship of war) on their late ward, whereby his resources were yet further straitened. He now became a professional writer of speeches or pleas for the law-courts, sometimes speaking himself. Biographers have delighted to relate how painfully Demosthenes made himself a tolerable speaker,-how, with pebbles in his mouth, he tried his lungs against the waves, how he declaimed as he ran up hill, how he shut himself up in a cell, having first guarded himself against a longing for the haunts of men by shaving one side of his head, how he wrote out Thucydides eight times, how he was derided by the Assembly and encouraged by a judicious actor who met him moping about the Peiræus. He certainly seems to have been the reverse of athletic (the stalwart Eschines upbraids him with never having been a sportsman), and he probably had some sort of defect or impediment in his speech as a boy. Perhaps the most interesting fact about his work for the law-courts is that he seems to have continued it, in some measure, through the most exciting parts of his great political career. The speech for Phormio belongs to the same year as the plea for Megalopolis. The speech for Bootus "Concerning the Name" comes between the First Philippic and the First Olynthiac. The speech against Pantænetus comes between the speech "On the Peace" and the Second Philippic.

In Jebb's Attic Orators from Antiphon to Isaeos, vol. ii. p. 267 ƒ., the traditions of the relation between Demosthenes and Isæus are examined in detail. It is there shown that the intercourse of the men can scarcely have been either intimate or prolonged, but that Demosthenes undoubtedly learned from Isæus the art of grappling with a forensic adversary in close and strenuous argument,